Many graph-theoretic problems that are NP-hard on general instances admit polynomial-time algorithms on graph families whose treewidth is sufficiently slowly growing as a function of the number of vertices [1]}. There is vast literature concerned with finding methods to relate the treewidth of graphs to other well-studied combinatorial parameters and leveraging this to devise efficient algorithms for algorithmic problems in graphs with constant or logarithmic treewidth [2]}. An excellent introduction to the concept of treewidth as well as brief survey of the work of Robertson and Seymour in establishing this concept can be found in Chapter 12 of [3]}.
